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A refinement of the stable pair invariants of Pandharipande and Thomas fornon-compact Calabi-Yau spaces is introduced based on a virtualBialynicki-Birula decomposition with respect to a C* action on the stable pairmoduli space, or alternatively the equivariant index of Nekrasov and Okounkov.This effectively calculates the refined index for M-theory reduced on theseCalabi-Yau geometries. Based on physical expectations we propose a productformula for the refined invariants extending the motivic product formula ofMorrison, Mozgovoy, Nagao, and Szendroi for local P^1. We explicitly computerefined invariants in low degree for local P^2 and local P^1 x P^1 and checkthat they agree with the predictions of the direct integration of thegeneralized holomorphic anomaly and with the product formula. The modularity ofthe expressions obtained in the direct integration approach allows us to relatethe generating function of refined PT invariants on appropriate geometries toNekrasov's partition function and a refinement of Chern-Simons theory on a lensspace. We also relate our product formula to wallcrossing.

The refined BPS index from stable pair invariants